Decoherence scaling transition in the dynamics of quantum information scrambling

Reliable processing of quantum information for developing quantum technologies requires precise control of out-of-equilibrium many-body systems. This is a highly challenging task because the fragility of quantum states to external perturbations increases with the system size. Here, we report on a series of experimental quantum simulations that quantify the sensitivity of a controlled Hamiltonian evolution to perturbations that drive the system away from the targeted evolution. Based on out-of-time ordered correlations, we demonstrate that the decay rate of the process fidelity increases with the effective number $K$ of correlated qubits as $K^{\alpha }$. As a function of the perturbation strength, we observe a decoherence scaling transition of the exponent $\alpha$ between two distinct dynamical regimes. In the limiting case below the critical perturbation strength, the exponent $\alpha$ drops sharply below 1, and there is no inherent limit to the number of qubits that can be controlled. This resilient quantum feature of the controlled dynamics of quantum information is promising for reliable control of large quantum systems.

Figure 1

Time evolution of the controlled-dynamics' fidelity and the corresponding effective cluster size of correlated spins as a measure of the scrambling of information. (a) The fidelity decay $f\left(t\right)=\text{Tr}\left[I_z\left(t\right)I_z^0\left(t\right)\right]/\mathrm{Tr}\left(I_z^2\right)$ is shown for two perturbation strengths for the perturbation Hamiltonian $\mathrm{\Sigma }={\mathcal{H}}_{dd}$. The strongest perturbation shows an exponential decay law for times $>0.3$ ms (inset). A MQC-fidelity $f_M\left(t\right)$ between the perturbed dynamics $I_z\left(t\right)$ and the ideal—nonperturbed—dynamics $I_z^0\left(t\right)$ as a function of the coherence order $M$. The enclosed area gives the global fidelity $f\left(t\right)$. (b) Evolution of the cluster size of correlated spins $K\left(t\right)$ determined from the second moment of the MQC fidelity (inset). The number $K\left(t\right)$ of correlated spins defines the “coherence length” on which the density matrices are comparable. For the weakest perturbations, the cluster size grows indefinitely, while for the strongest ones, $K\left(t\right)$ reaches a stationary value—an effect we call localization. (c) The instantaneous decoherence rate ${\chi }^{\prime }(p,t)=\frac{d\chi }{dt}(p,t)$ of the fidelity $f\left(t\right)$ as a function of time $t$. The exponential decay regimen of $f\left(t\right)$ is manifested here when the decoherence rate ${\chi }^{\prime }\left(t\right)$ achieves a constant value. (d) The instantaneous decoherence rate ${\chi }^{\prime }$ as a function of the cluster size $K$. The plateau of ${\chi }^{\prime }\left(t\right)$ that appears when the cluster size $K\left(t\right)$ localizes in panel (c), here is manifested by the accumulation of points at the end of the curve ${\chi }^{\prime }\left(K\right)$.

Figure 2

Instantaneous decoherence rate ${\chi }^{\prime }$ as a function of the effective cluster size $K$. The two perturbation Hamiltonians are considered: (a) $\mathrm{\Sigma }={\mathcal{H}}_{dd}$ and (b) $\mathrm{\Sigma }=H_z$. At long times, the fidelity decay rate is driven by the scrambling rate ${\chi }^{\prime }\left(K\right)\sim K^{\alpha }$ given by the instantaneous cluster-size, with a power-law exponent that depends on the perturbation strength $p$. (c) The power-law exponent $\alpha$ decreases with decreasing the perturbations strength, showing two plateau values at the weakest (${\alpha }_0$) and at strongest perturbation (${\alpha }_{\infty }$). To estimate ${\alpha }_0$ and ${\alpha }_{\infty }$, we fit $\alpha \left(p\right)$ with a sigmoid function (solid line).

Figure 3

Decoherence scaling transition between two dynamical regimes of the fidelity decay evinced by the finite-time scaling analysis. Scalings for both perturbation Hamiltonians (a) $\mathrm{\Sigma }={\mathcal{H}}_{dd}$ and (b) $\mathrm{\Sigma }={\mathcal{H}}_z$ are shown. (c) The corresponding scaling factors $\zeta \left(p\right)$ and their fittings to the function $\zeta \left(p\right)=A|p-p_c{|}^{-\nu }+B$. In the case of $\mathrm{\Sigma }={\mathcal{H}}_{dd}$, the critical exponents are $\nu =(-0.57\pm 0.03)$ and $s=(-0.93\pm 0.06)$ and the critical perturbation $p_c=(0.026\pm 0.001)$. For $\mathrm{\Sigma }={\mathcal{H}}_z$, the critical perturbation is $p_c=(0.065\pm 0.01)$, and the critical exponents are $\nu =(-0.47\pm 0.05)$ and $s=(-0.93\pm 0.06)$. The curves of $\zeta \left(p\right)$ are normalized to satisfy $\zeta \left(p_{\infty }\right)={\left[{\chi }^{\prime }\left(K\right)/K^{{\alpha }_{\infty }}\right]}^{1/2}$, where $p_{\infty }=0.108$ for $\mathrm{\Sigma }={\mathcal{H}}_{dd}$ and $p_{\infty }=0.24$ for $\mathrm{\Sigma }={\mathcal{H}}_z$ are the largest perturbation strength used in the experiments.